Solve (x-1)(x+1)(x+2)(x-1/2)

$$(x-1)(x+1)(x+2)(x- \frac{ 1 }{ 2 } )$$

Evaluate

$$\frac{\left(2x-1\right)\left(x+2\right)\left(x^{2}-1\right)}{2}$$

Solution Steps

Apply the distributive property by multiplying each term of x-1 by each term of x+1.

$$\left(x^{2}+x-x-1\right)\left(x+2\right)\left(x-\frac{1}{2}\right)$$

Combine x and -x to get 0.

$$\left(x^{2}-1\right)\left(x+2\right)\left(x-\frac{1}{2}\right)$$

Apply the distributive property by multiplying each term of x^{2}-1 by each term of x+2.

$$\left(x^{3}+2x^{2}-x-2\right)\left(x-\frac{1}{2}\right)$$

Apply the distributive property by multiplying each term of x^{3}+2x^{2}-x-2 by each term of x-\frac{1}{2}.

$$x^{4}+x^{3}\left(-\frac{1}{2}\right)+2x^{3}+2x^{2}\left(-\frac{1}{2}\right)-x^{2}-x\left(-\frac{1}{2}\right)-2x-2\left(-\frac{1}{2}\right)$$

Combine x^{3}\left(-\frac{1}{2}\right) and 2x^{3} to get \frac{3}{2}x^{3}.

$$x^{4}+\frac{3}{2}x^{3}+2x^{2}\left(-\frac{1}{2}\right)-x^{2}-x\left(-\frac{1}{2}\right)-2x-2\left(-\frac{1}{2}\right)$$

Cancel out 2 and 2.

$$x^{4}+\frac{3}{2}x^{3}-x^{2}-x^{2}-x\left(-\frac{1}{2}\right)-2x-2\left(-\frac{1}{2}\right)$$

Combine -x^{2} and -x^{2} to get -2x^{2}.

$$x^{4}+\frac{3}{2}x^{3}-2x^{2}-x\left(-\frac{1}{2}\right)-2x-2\left(-\frac{1}{2}\right)$$

Multiply -1 and -\frac{1}{2} to get \frac{1}{2}.

$$x^{4}+\frac{3}{2}x^{3}-2x^{2}+\frac{1}{2}x-2x-2\left(-\frac{1}{2}\right)$$

Combine \frac{1}{2}x and -2x to get -\frac{3}{2}x.

$$x^{4}+\frac{3}{2}x^{3}-2x^{2}-\frac{3}{2}x-2\left(-\frac{1}{2}\right)$$

Multiply -2 times -\frac{1}{2}.

$$x^{4}+\frac{3}{2}x^{3}-2x^{2}-\frac{3}{2}x+1$$

Expand

$$x^{4}+\frac{3x^{3}}{2}-2x^{2}-\frac{3x}{2}+1$$

Solution Steps

Apply the distributive property by multiplying each term of x-1 by each term of x+1.

$$\left(x^{2}+x-x-1\right)\left(x+2\right)\left(x-\frac{1}{2}\right)$$

Combine x and -x to get 0.

$$\left(x^{2}-1\right)\left(x+2\right)\left(x-\frac{1}{2}\right)$$

Apply the distributive property by multiplying each term of x^{2}-1 by each term of x+2.

$$\left(x^{3}+2x^{2}-x-2\right)\left(x-\frac{1}{2}\right)$$

Apply the distributive property by multiplying each term of x^{3}+2x^{2}-x-2 by each term of x-\frac{1}{2}.

$$x^{4}+x^{3}\left(-\frac{1}{2}\right)+2x^{3}+2x^{2}\left(-\frac{1}{2}\right)-x^{2}-x\left(-\frac{1}{2}\right)-2x-2\left(-\frac{1}{2}\right)$$

Combine x^{3}\left(-\frac{1}{2}\right) and 2x^{3} to get \frac{3}{2}x^{3}.

$$x^{4}+\frac{3}{2}x^{3}+2x^{2}\left(-\frac{1}{2}\right)-x^{2}-x\left(-\frac{1}{2}\right)-2x-2\left(-\frac{1}{2}\right)$$

Cancel out 2 and 2.

$$x^{4}+\frac{3}{2}x^{3}-x^{2}-x^{2}-x\left(-\frac{1}{2}\right)-2x-2\left(-\frac{1}{2}\right)$$

Combine -x^{2} and -x^{2} to get -2x^{2}.

$$x^{4}+\frac{3}{2}x^{3}-2x^{2}-x\left(-\frac{1}{2}\right)-2x-2\left(-\frac{1}{2}\right)$$

Multiply -1 and -\frac{1}{2} to get \frac{1}{2}.

$$x^{4}+\frac{3}{2}x^{3}-2x^{2}+\frac{1}{2}x-2x-2\left(-\frac{1}{2}\right)$$

Combine \frac{1}{2}x and -2x to get -\frac{3}{2}x.

$$x^{4}+\frac{3}{2}x^{3}-2x^{2}-\frac{3}{2}x-2\left(-\frac{1}{2}\right)$$

Multiply -2 times -\frac{1}{2}.

$$x^{4}+\frac{3}{2}x^{3}-2x^{2}-\frac{3}{2}x+1$$